What is the box that appears around this mesh in edit mode?
What you're seeing is the original mesh - the cube. In the tutorial they add a Subsurf modifier and the cube immediately turns into a low poly sphere. However, the modifier has not been applied to the cube, meaning the original cube geometry still exists, which is what you see when they enter Edit Mode. 

This allows you to move the original geometry of the cube and see the Subsurf result update in real-time. The original geometry is called the Edit Cage when there is a Subsurf modifier present.

Here I have added a Subsurf modifier to Blender's 'Monkey' mesh. You can see the original, low-poly cage (the black wires) around the smooth Subsurf geometry (the grey surface):



So, in short, it was simply changing to edit mode that made the box (the original geometry) appear.

You can read more about modifiers, how their results are displayed in the viewport and their visibility options, here.